14 Nonlinear 3D Dynamic Model of an Automotive Dual Mass Flywheel 133 Modelling the dynamic behaviour of DMF is of utter importance to accurately reproduce and study torsional vibrations and rattle phenomenon affecting vehicle driveline and consequently several models were proposed in the technical literature. A simplified 2 degrees of freedom (dofs) dynamic model of DMF is presented by Walter et al. [1, 2] and by Lei et al. in [3]. The model consists of two rotating inertias connected by a linear spring-damper element. In [4], a 1D lumped-parameters nonlinear dynamic model of DMF is proposed, describing friction phenomena occurring between arc springs and primary mass due to centrifugal and redirection forces, which provide a physical explanation of hysteresis and springs hardening. In [5], Lei et al. updated the previously developed 2-dofs model introducing a nonlinear spring connecting the two rotatory inertias and a Bouc–Wen model to introduce the hysteretic frictional behaviour of DMF. Beside the great effort spent to investigate and reproduce the torsional behaviour of DMF, a model capable of studying its 3D dynamics and especially able of estimating the radial forces transmitted by DMF to vehicle powertrain has still to be developed. The present paper proposes a 3D nonlinear model of DMF based on MultyBody (MB) approach. Great attention is given to modelling contacts between arc springs and DMF stages so to evaluate radial forces transmitted to the powertrain. First, capability of the developed model to reproduce DMF behaviour was assessed by means of static and dynamic experimental tests. Then the DMF model was introduced into a complete MB model of the vehicle powertrain to evaluate its influence on powertrain modes of vibration. 14.2 DMF Torsional Model In this section, the multibody model of DMF is presented. In the following subsections, DMF is modelled allowing only rotations of primary and secondary masses around DMF axis (torsional model). This model allows to investigate the torsional behaviour of DMF and the comparison with the experimental data. In Sect. 14.4, constrains of primary and secondary masses will be removed so to allow relative rotations and translations of primary and secondary masses (3D model). In this way, the effects of 3D dynamics of DMF on vehicle powertrain will be evaluated. The model of the DMF is implemented using a commercial MB code. 14.2.1 Description of Bodies, Constraints and Number of d.o.f. In the developed MB model of DMF primary mass and secondary mass are considered as rigid bodies. The shape of these bodies is carved in order to create two or more (three in the present application) sections where the arc springs are housed. Sections hosting arc springs are bounded by stoppers equally spaced along the primitive circumference of primary and secondary masses (Fig. 14.2a). Generally speaking, stopper profiles are the same for every sector but they can also be slightly different due to manufacturing tolerance or design choice. Concerning arc springs, we consider the most generic case with two concentric arc springs for each sector. Inner and outer arc springs have different angular extension, in this way it is possible to obtain a different DMF torsional stiffness at different relative angular displacement between primary and secondary masses. Note that the centrifugal force pushes the arc springs against the inner surface of the primary mass as angular speed increases. To include centrifugal force effects into the model, each arc spring is modelled as a series of lumped springs and masses [6, 7], as shown in Fig. 14.2b. In the case of the torsional model, primary and secondary masses are constrained to the ground with revolute joints. Outer arc spring lumped masses are constrained to the primary DMF mass with revolute joints. Inner arc spring lumped masses are constrained to the outer arc spring elements with a revolute joint. Assuming to divide each of the three arc springs in 6 elements (Fig. 14.2b), the torsional model has thus 44 dofs: rotations of primary and secondary masses and 42 dofs related to the lumped masses schematizing the arc springs.
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